Demystifying Zeno’s Paradox: Achilles and the Tortoise

Demystifying Zeno's Paradox Achilles and the Tortoise "In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead." as recounted by Aristotle* Description of the Paradox Achilles is in a footrace with the tortoise. Achilles allows the tortoise a head start of 100 meters. Assuming each racer starts running at some constant speed (10m/s for Achilles and 1m/s for the tortoise to simplify), then after some finite time (10s), Achilles will have run 100m, bringing him to the tortoise's starting point. During this time, the tortoise has run a much shorter distance (10m). It will then take Achilles some further time (1s) to run that distance, by which time the tortoise will have advanced farther (o.1m); and then more time still to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles reaches somewhere the tortoise has been, he still has farther to go. Therefore, because there are an infinite number of points he must reach where the tortoise has already been, Achilles can never overtake the tortoise. Our Explanation In reality, this is what happens in the first 12 seconds using simple physics and math: ** To solve for the point when and where Achilles reaches the tortoise: D,=10T, and D,=T,+100 runs at 10 m/s runs at 1 m/s T=11.1111111111111111111... seconds D=111.111111111111111111... meters A. Achilles t%3DOS t=105 t=125 overtakes the tortoise 100 110 112 120 m The thing is, the way the paradox is originally presented is like watching a recording of the race, advancing it frame by frame. In each new frame, Achilles reaches where the tortoise was in the preceding frame, while the tortoise has moved slightly ahead. We explore the highlighted part of the race using these time frames. B. t=10s t=11.15 t=11.1115 t=115 t=11.115 t=11.1111s and so on 100 111.11 111.1111 m 110 111.1 111.111 In each new frame, the time increment becomes smaller and smaller, approaching zero at T, (when Achilles actually reaches the tortoise), thus creating infinite steps towards that point. Another way to think of it is like watching the race in slow motion and slowing it down further and further until you come to a complete halt at T,. You WILL NOT OBSERVE Achilles reaching the tortoise, because you choose not to get to that point, but that doesn't mean that it doesn't happen! By describing the race this way, Zeno creates infinite steps to a finite point in time (T.). He uses the term "the quickest runner can NEVER overtake the slowest" while ignoring the normal flow of time. Our Point: Do not let 3,000 year old philosophy get in the way of simple physics, math and common sense. * http://en.wikipedia.org/wiki/Zeno's_paradoxes ** Graphs are not to scale in order to simplify the visual presentation Formore details: http://www.clickexist.com/2013/05/25/achilles-tortoise-paradox CE ClickExist.com